Properties

Degree 2
Conductor $ 2 \cdot 3 \cdot 79 \cdot 211 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 2

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 3-s + 4-s − 3·5-s − 6-s − 4·7-s + 8-s + 9-s − 3·10-s − 12-s − 2·13-s − 4·14-s + 3·15-s + 16-s + 18-s − 5·19-s − 3·20-s + 4·21-s − 2·23-s − 24-s + 4·25-s − 2·26-s − 27-s − 4·28-s − 2·29-s + 3·30-s + 3·31-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 1/2·4-s − 1.34·5-s − 0.408·6-s − 1.51·7-s + 0.353·8-s + 1/3·9-s − 0.948·10-s − 0.288·12-s − 0.554·13-s − 1.06·14-s + 0.774·15-s + 1/4·16-s + 0.235·18-s − 1.14·19-s − 0.670·20-s + 0.872·21-s − 0.417·23-s − 0.204·24-s + 4/5·25-s − 0.392·26-s − 0.192·27-s − 0.755·28-s − 0.371·29-s + 0.547·30-s + 0.538·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 100014 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100014 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(100014\)    =    \(2 \cdot 3 \cdot 79 \cdot 211\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{100014} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  2
Selberg data  =  $(2,\ 100014,\ (\ :1/2),\ 1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;79,\;211\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;79,\;211\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 + T \)
79 \( 1 + T \)
211 \( 1 + T \)
good5 \( 1 + 3 T + p T^{2} \)
7 \( 1 + 4 T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 + 5 T + p T^{2} \)
23 \( 1 + 2 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 - 3 T + p T^{2} \)
37 \( 1 + 7 T + p T^{2} \)
41 \( 1 - 12 T + p T^{2} \)
43 \( 1 + 6 T + p T^{2} \)
47 \( 1 - 2 T + p T^{2} \)
53 \( 1 - 4 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 + p T^{2} \)
67 \( 1 + 10 T + p T^{2} \)
71 \( 1 + 10 T + p T^{2} \)
73 \( 1 + 9 T + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
89 \( 1 - 15 T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−14.36784119557231, −13.47216846565329, −13.22468151665680, −12.70470968419756, −12.22211317712797, −11.97293580044803, −11.53394115241960, −10.82826740933800, −10.42178642355420, −10.04743048981858, −9.282414957195946, −8.835105854609459, −8.088503461376283, −7.587006937837323, −7.047613803850972, −6.699542224063991, −6.057638255084706, −5.731440296788325, −4.860866810676057, −4.400681216597361, −3.861141141620363, −3.491344342071186, −2.773616152125252, −2.209040771998031, −1.117593748734256, 0, 0, 1.117593748734256, 2.209040771998031, 2.773616152125252, 3.491344342071186, 3.861141141620363, 4.400681216597361, 4.860866810676057, 5.731440296788325, 6.057638255084706, 6.699542224063991, 7.047613803850972, 7.587006937837323, 8.088503461376283, 8.835105854609459, 9.282414957195946, 10.04743048981858, 10.42178642355420, 10.82826740933800, 11.53394115241960, 11.97293580044803, 12.22211317712797, 12.70470968419756, 13.22468151665680, 13.47216846565329, 14.36784119557231

Graph of the $Z$-function along the critical line